What is the integral of ((4x-7))/(x^2-6x+13) ?

The integral of ((4x-7))/(x^2-6x+13) is 5/2 arctan(1/2 (x-3))+2ln|x^2-6x+13|+C

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integral of ((4x-7))/(x^2-6x+13)

\int \frac{( 4 x - 7 )}{x ^{2} - 6 x + 13} d x

\frac{5}{2} arctan (\frac{1}{2} (x - 3)) + 2 ln x^{2} - 6 x + 13 + C

Solution steps

\int \frac{4 x - 7}{x ^{2} - 6 x + 13} d x

Apply Integration By Parts

= \frac{1}{2} arctan (\frac{1}{2} (x - 3)) (4 x - 7) - \int 2 arctan (\frac{1}{2} (x - 3)) d x

\int 2 arctan (\frac{1}{2} (x - 3)) d x = 2 (x arctan (\frac{1}{2} (x - 3)) - ln x^{2} - 6 x + 13 - 3 arctan (\frac{1}{2} (x - 3)))

= \frac{1}{2} arctan (\frac{1}{2} (x - 3)) (4 x - 7) - 2 (x arctan (\frac{1}{2} (x - 3)) - ln x^{2} - 6 x + 13 - 3 arctan (\frac{1}{2} (x - 3)))

Simplify

\frac{1}{2} arctan (\frac{1}{2} (x - 3)) (4 x - 7) - 2 (x arctan (\frac{1}{2} (x - 3)) - ln x^{2} - 6 x + 13 - 3 arctan (\frac{1}{2} (x - 3))) : \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + 2 ln x^{2} - 6 x + 13

= \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + 2 ln x^{2} - 6 x + 13

Add a constant to the solution

= \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + 2 ln x^{2} - 6 x + 13 + C

What is the integral of ((4x-7))/(x^2-6x+13) ?
The integral of ((4x-7))/(x^2-6x+13) is 5/2 arctan(1/2 (x-3))+2ln|x^2-6x+13|+C